We consider complex algebraic varieties with a linear group action. The basic example is the homogeneous space $G/P$, where $G$ is a reductive algebraic group and $P$ is a parabolic subgroup. It contains Schubert varieties, which are the closures of the orbits of the Borel subgroup $B$. In general the Schubert varieties are singular. The Schubert varieties are invariant with respect to the maximal torus action. In such a situation (a singular $T$-invariant subset in a smooth ambient space $M$) we study various characteristic classes:

- the Chern-Schwartz-MacPherson classes in equivariant cohomology of $M$

- the motivic Chern class in equivariant K-theory

- a version of Borisov-Libgober elliptic class living in equivariant elliptic cohomology

We give a set of axioms allowing to compute the mentioned classes. The axioms are modeled on the properties of Okounkov stable envelopes. In the case $M=G/B$, we discuss an action of Hecke-type algebra which reproduces characteristic classes. Formulas in cohomology and K-theory were already obtained by Aluffi-Mihalcea-Schurmann-Su, and we extend their results to elliptic classes, also for the case of $G/P$. The inductive formulas are generalizations the classical results of Demazure, Bernstein-Gelfand-Gelfand, Lusztig and Lascoux-Schutzenberger.

This is a joint work with Laszlo Feher, Richard Rimanyi (and with Shrawan Kumar for the elliptic case).